A Primer on Q-Curvature

by Michael Eastwood and Jan Slovák

Disclaimer: These are rough notes only, aimed at setting the scene and
promoting discussion at the American Institute of Mathematics Research
Conference Center Workshop `Conformal Structure in Geometry,
Analysis, and Physics,'
-
August 2003.
For simplicity, we have omitted all references. Curvature conventions are in an
appendix. Conversations with Tom Branson and Rod Gover have been extremely
useful.

Let be an oriented even-dimensional Riemannian -manifold. Branson's
-curvature is a canonically defined -form on . It is not conformally
invariant but enjoys certain natural properties with respect to conformal
transformations.

When , the -curvature is a multiple of the scalar curvature. With
conventions as in the appendix . Under conformal rescaling of the
metric,
we have

where
is the Laplacian.

When , the -curvature is given by

(1)

Under conformal rescaling,

where is the Paneitz operator

(2)

For general even , the -curvature transforms as follows:-

(3)

where is a linear differential operator from functions to -forms whose
symbol is . It follows from this transformation law that is
conformally invariant. To see this, suppose that

Then

but also

Therefore,
. With suitable normalisation, is
the celebrated Graham-Jenne-Mason-Sparling operator. Thus, may be regarded
as more primitive than and, therefore, is at least as mysterious.

Even when is conformally flat, the existence of is quite subtle. We can
reason as follows. When is actually flat then must vanish. Therefore,
in the conformally flat case, locally if we write
where is flat, then () implies that

(4)

where is the ordinary Laplacian in Euclidean space with as
metric. An immediate problem is to verify that this purported construction of
is well-defined. The problem is that there is some freedom in writing
as proportional to a flat metric. If
also
, then we must show that

This easily reduces to two facts:-

fact 1:

is conformally invariant on flat space.

fact 2:

if is itself flat, then
.

The second of these is clearly necessary in order that () be
well-defined. For it is immediate from (). For it
may be verified by direct calculation as follows. If and
are both flat then

(5)

where
. Therefore,

and

whence

(6)

Taking the trace of () gives

and now () gives, by induction,

In particular,
, as advertised.

That is conformally invariant on flat space is well-known. It
may also be verified directly by a rather similar calculation. For example,
here is the calculation when . For general conformally related metrics
in dimension 4,

If is flat then the third order terms cancel leaving

If
is also flat, then () implies

whence the second order terms cancel and the first order ones simplify:-

But using () again,

and the first order terms also cancel leaving
, as
advertised.

Conundrum: Deduce fact 2 from fact 1 or vice versa. Both are consequences
of (). Alternatively, construct a Lie algebraic proof of fact 2. There is a Lie algebraic proof of fact 1. It corresponds to the
existence of a homomorphism between certain generalised Verma modules for
.

What about a formula for , even in the conformally flat case? We have a
recipe for , namely (), but it is not a formula. We may
proceed as follows.

If
and is flat, then
() implies that

(7)

Taking the trace yields

(8)

This identity is also valid when : it is (). Dropping the hat
gives
. This is the simplest of the desired formulae.

To proceed further we need two identities. If has conformal weight ,
then as described in the appendix,

which we rewrite as

(9)

Similarly, if has weight , then

(10)

and, if is symmetric and has weight , then

(11)

The quantities in () have weight . Therefore, applying
() gives

wherein we may use () to replace
to
obtain

We may now apply , using (), (), and
() to replace by
on the right hand
side and () to replace derivatives of . We obtain
an expression involving only complete contractions of
, its
hatted derivatives, and :-

Using the Bianchi identity
, we may
rewrite this as

(12)

and, in particular, conclude that when ,

(13)

Though it is only guaranteed that this formula is valid in the conformally flat
case, in fact it agrees with the general expression () in dimension 4.

Of course, we may continue in the vein, further differentiating () to
obtain a formula for
expressed in terms of complete
contractions of
, its hatted derivatives, and .
With increasing , this gets rapidly out of hand. Moreover, it is only
guaranteed to give in the conformally flat case. Indeed, when this
naive derivation of fails for a general metric. Nevertheless, there
are already some questions in the conformally flat case.

Conundrum: Find a formula for in the conformally flat case. Show that the
procedure outlined above produces a formula for .

In fact, there is a tractor formula for the conformally flat . This is not
the place to explain the tractor calculus but, for those who know it already:-

where

Unfortunately, this formula hides a lot of detail and does not seem to be of
much immediate use. It is not valid in the curved case.

Recall that, like , the Pfaffian is an -form canonically associated to a
Riemannian metric on an oriented manifold in even dimensions. It is defined as
a complete contraction of copies of the Riemann tensor with two copies of
the volume form. For example, in dimension four it is

where
is the volume form normalised, for example, so that

Therefore, in four dimensions,

The integral of the Pfaffian on a compact manifold is a multiple of the Euler
characteristic. In dimension 4, for example,

Notice the simple relationship between and in dimension 4:-

Of course, it follows from () that is a conformal
invariant. Also, in the conformally flat case, it follows from a theorem of
Branson, Gilkey, and Pohjanpelto that must be a multiple of the Pfaffian
plus a divergence. However, the link between and the Pfaffian is extremely
mysterious.

Conundrum: Find a direct link between and the Pfaffian in the conformally flat
case. Prove directly that is a topological invariant in this case.

Conundrum: Is it true that, on a general Riemannian manifold, may be written as a
multiple of the Pfaffian plus a local conformal invariant plus a divergence?

Recall the conventions for Weyl structures as in the appendix. In particular, a
metric in the conformal class determines a -form . In fact, a
Weyl structure may be regarded as a pair
subject to
equivalence under the simultaneous replacements

A Riemannian structure induces a Weyl structure by taking the equivalence class
with but not all Weyl structures arise in this way. A Weyl
structure gives rise to a conformal structure by discarding . We may
ask how -curvature is related to Weyl structures. From the transformation
property (), it follows that may be defined for a Weyl

structure as follows. Since is a Riemannian invariant, the differential
operator is necessarily of the form
for some
Riemannian invariant linear differential operator from -forms to -forms.
Now, if
is a Weyl structure, choose a representative metric
and consider the -form

where is the Riemannian -curvature associated to and
is the -form associated to . If
,
then

(14)

In dimension 4 we can proceed further as follows. From () we see
that

and so we may calculate

In combination with () we obtain

However,

and, therefore,

(15)

is an invariant of the Weyl structure that agrees with when the Weyl
structure arises from a Riemannian structure.

Conundrum: Can we find such a in general even dimensions?
Presumably, this would restrict the choice of Riemannian .

Though given by () is an invariant of the Weyl
structure, it is not manifestly so. Better is to rewrite it as follows. Using
conventions from the appendix, we may write the Schouten tensor
() of the Weyl structure in terms of the Schouten tensor of a
representative metric :-

In particular,

Therefore, recalling the formula () for in dimension 4,

whence, from (),

However,

and so

a manifest invariant of the Weyl structure, as required.

Conundrum: Did we really need to go through this detailed calculation? What are the
implications, if any, for the operator
?

Conundrum: Can we characterise the Riemannian by sufficiently many properties? Do
Weyl structures help in this regard?

Tom Branson has suggested that, for two metrics and
in the same conformal class on a compact manifold , one should consider the
quantity

That it is a cocycle,

is easily seen to be equivalent to the GJMS operators being self-adjoint.

Conundrum: Are there any deeper properties of Branson's cocycle
?

One possible rôle for is in a curvature prescription problem:-

Conundrum: On a given manifold , can one find a metric with specified ?

One can also ask this question within a given conformal class or within the
realm of conformally flat metrics though, of course, if is compact, then
must be as specified by the conformal class and the topology of .
There is also the question of uniqueness:-

Conundrum: When does determine the metric up to constant rescaling within a given
conformal class?

Since we know how changes under conformal rescaling (),
this question is equivalent to

Conundrum: When does the equation have only constant solutions?

On a compact manifold in two dimensions this is always true: harmonic functions
are constant. In four dimensions, though there are conditions under which
has only constant solutions, there are also counterexamples, even on
conformally flat manifolds. The following counterexample is due to Michael
Singer and the first author. Consider the metric in local coördinates

It is easily verified that it is conformally flat, scalar flat, and has

From () we see that if is a function of alone, then
, where is the Laplacian for the two-dimensional metric

More specifically, in these local coördinates

It is easily verified that annihilates the following functions:-

In fact, are stereographic coördinates on the sphere and these three
functions extend to the sphere to span the spherical harmonics of minimal
non-zero energy. On then other hand, the metric

is the hyperbolic metric on the disc. We conclude that the Paneitz operator has
at least a 4-dimensional kernel on . The same conclusion
applies to
where is any Riemann surface of genus
equipped with constant curvature metric as a quotient of . (In
fact, the dimension in this case is exactly 4.)

APPENDIX: Curvature Conventions

Firstly, our conventions for conformal weight. A density of conformal
weight may be identified as a function for any metric in the conformal
class. At the risk of confusion, we shall also write this function as . If
however, our choice of metric is replaced by a conformally equivalent
, then the function is replaced by
. Quantities that are not conformally invariant can still
have a conformal weight with respect to constant rescalings. For example, the
scalar curvature has weight in this respect. Explicit conformal
rescalings are generally suppressed.

The Riemann curvature is defined by

The Ricci and scalar curvatures are

respectively. The Schouten tensor is

and transforms under conformal rescaling by

(16)

In particular, if
are two flat metrics,
then

a tensor version of the Riccati equation. When , the Schouten tensor
itself is not defined but its trace is well-defined:-

(17)

and so, if
are two flat metrics, then
.

A Weyl structure is a conformal structure together with a choice of
torsion-free connection preserving the conformal structure. In other
words, if we choose a metric in the conformal class, then

determining a smooth -form . Conversely,
determines :

where is the Levi-Civita connection for the metric . Let
denote the Ricci curvature of the connection :-

We may compute these curvatures in terms of and , for
a chosen metric in the conformal class:-

so

whence

and

whose trace is

Therefore,

If two Weyl structures have the same underlying conformal structure, then we
may, without loss of generality, represent them as
and
for the same metric and an arbitrary
-form . If we write hatted quantities to denote those computed
with respect to
, then

for

(18)

we have the convenient transformation law

(19)

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